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In Unit 2, you revisited different ways to label a triangle. You recognized that the choice of labelling the triangle depended on what you would be doing with the information.
 

This is a picture of 2 identical triangles. The first triangle has only the vertices labeled with the uppercase letters D, E, and F.  The angle at vertex E is marked as a ninety degree angle. The second triangle has the vertices labeled the same way, but also shows the labelling of the sides using lower case letters, d, e, and f.

 

You also learned to refer to side lengths of a right triangle with respect to a reference angle.  This method of naming the side lengths is used in writing the definitions of the primary trigonometric ratios.

​  This is a picture of 2 write triangles, both labelled as triangle ABC. The fiirst triangle illustrates naming the sides (opposite, adjacent, and hypotenuse) of the triangle where ∠A is the reference angle. Side AB is labelled as “adjacent to A” and side CB is labelled as “opposite angle A”. Side CA is labelled as “hypotenuse”. The second triangle illustrates naming the sides of the triangle where ∠C is the reference angle. Side CB is labelled as “adjacent to C” and side AB is labelled as “opposite angle C”. Side CA is labelled as “hypotenuse”.

When you create and label a drawing or diagram, you take the given information and add it to the diagram. When you transfer the information from the diagram into an  equation using the primary trigonometric ratios, you need to refer to the sides using  the reference angle in the diagram.

Practice Questions:

triangleQuiz

Long Description

 

This is the discussion icon.Considering the Steps

In the quiz above, you read a description of a triangle and then found a drawing that matched the description. What did you do as you were inspecting and then selecting the correct drawing? What did you look at first? And then what did you consider?

You will be creating and labelling triangles in order to solve for an unknown side or angle.

Although each person’s sketch may look different, it is important to start with a reasonable representation of the information. This will help you choose the appropriate trigonometric ratio and accurately substitute values into your equation so that you will determine the unknown value correctly.

Think about the steps that you would follow to transfer the written description of a triangle to an accurately labelled drawing. What would you do first? Next?

Save these step to your Portfolio.

 
 
Action.

ACTION

This is the Portfolio icon. Considering the Steps

In this activity, the Mathematical Process Connecting the focus.

Open your document U3 Mathematical Processes and read the descriptions for the process.

As you complete the activity, notice when you are making connections between the new topic and the previous work that you have done.

Insert your record below the description of the process.

 

In this activity, you will be reading descriptions of situations that can be modelled with triangles. You will create and label a sketch that will help to set up the equation in order to solve for an unknown side or angle.

triangles

Long Description

You have been able to successfully transfer the description of a triangle from words into a diagram. You were then able to solve for the unknown side or angle using the diagram to create the correct equation for the situation.

Describing Angles in Right Triangle Problems

Trigonometry is often used to determine distances or angles in real life situations. Triangles may be used to model the problem but the description of the situation will not have labelled vertices. Heights of physical structures or the distance between them can be easily described. Specific vocabulary is used to describe angles from one point to another.

Angle of Elevation and Angle of Depression

To describe angles in a situation, the terms angle of elevation and angle of depression are commonly used. Both terms involve an invisible horizontal line that is created by standing straight and looking straight ahead.

This is an image that shows the position of the angle of elevation (looking up) and the position of the angle of depression (looking down).

If you look up to the top of a structure, you create an invisible line called the line of sight. The angle of elevation is the angle that your eyes move through from the invisible horizontal line to the invisible line of sight. Often, these two lines create 2 of the 3 sides of the triangle that you will use to model a problem.

If you are standing at a high point and look downwards towards an object, you create a line of sight. The angle of depression is the angle that you move your eyes through from the invisible horizontal line to the invisible line of sight. As described above, these two lines may be used to create two sides of the triangle that you will use to model a problem.

Problem 1

The angle of elevation from a sailboat to the top of a 130 m lighthouse on the shore measures 16°. To the nearest metre, how far is the sailboat from the shore?

An image of a lighthouse and a sail boat drawn to create a triangle.

The image shows the sailboat and the lighthouse. The invisible horizontal line is drawn at a level of someone’s eyes and the line of sight to the top of the lighthouse is added. These lines will create two sides of the triangle. The angle of elevation is shown as the angle between the two invisible lines.

The third side will be the height of the lighthouse. Notice that the horizontal line and the line for the height of the lighthouse meet at a 90° angle. In general, you make an assumption that vertical structures make a right angle with the ground.

With this information, you can draw a triangle, without the objects, to model the situation. This is followed by writing the equation to solve for the unknown value.

An iamge of the triangle created by the sailboat and lighthouse.

Letters at the vertices may be in reference to the object that was there - B for boat, G for ground and L for lighthouse. Choosing meaningful labels can help you remember the details of the problem.

You have values for one angle and the side opposite that angle, and you want to determine the length of the side that is adjacent to that angle. Label that side x, the unknown. Select the appropriate ratio.

The tangent ratio combines the angle, opposite and adjacent sides.

An image of how tangent of 16 degrees equals 130 over an unkown value can be solved to show that x = 453 m.

The sailboat is approximately 453 m from the shore.

Problem 2

A salvage ship is locating wreckage. The ship’s sonar (a device that uses sound waves to detect objects on the bottom of the sea) picks up a signal showing there is wreckage at an angle of depression of 12°. The sea charts for the region show an average depth of 40 m. If a diver is lowered directly below the salvage ship, how far can the diver expect to have to travel along the ocean floor to reach the wreckage? Round to the nearest metre.

An image of a salvage boat drawn at a 12 degree angle from wreckage underneath the water.

The image shows the salvage boat and the wreckage. The invisible horizontal line is drawn at the level of the boat, and the line of sight to the wreckage is added. The angle of depression is shown as the angle between the two invisible lines. Notice that if you create a triangle using the wreckage, the boat and the diver, the angle of depression would not be inside the triangle.

An image of the triangle created from the wreckage, the salvage ship and a ninety degree angle. The triangle is labeled DSW.

When you draw the triangle, the location of the wreckage, W, ship, S, and diver, D, create the three vertices. The diver is lowered to be directly below the ship and is assumed to be at a right angle to the floor of the ocean. The depth of the water is 40 m, so the length of SD is 40 m. At vertex S, there is a right angle so the angle within the triangle, WSD, can be calculated: 90° - 12° = 78°.

A triangle created by the wrckage, the salvage ship and a ninety degree angle. The 12 degree angle and the complimentary 78 degree angle have been added, as well as a label of 40m from the salvage ship to the bottom of the sea.

The three sides of the triangle are SD (the depth of the water, which is 40 m), SW (the sight line) and WD (the distance that the diver will travel along the floor of the ocean). ∠S in ΔSWD is 78° and is adjacent to SD and opposite WD. The tangent ratio ties together the opposite and adjacent sides with the reference angle.

Label the unknown distance, the length of WD, as x.

An image showing how the tangent of 78 degrees equals an unknown value over 40 can be solved to show that x equals 188 m.

The diver will need to travel 188 m along the ocean floor.

More Practice

For each question, draw and label a triangle to model the problem, then determine the value of the unknown length, to the nearest metre. 

  1. Marico has his own small plane. He is planning his approach to the local airport. He wants the angle of depression to be 3°. He is currently at an <> altitude (height that is at a right angle to the ground) of 700 m. What will be the distance he travels along his flight path as he descends to the airport?
  1. Qaiyaan is 1.5 m tall and is standing at the top of a 85 m cliff at the edge of a lake. Her friend, Sarii, is in a boat on the lake. From the boat, Sarii observes that the angle of elevation to the top of Qaiyaan’s head is 29°. Determine the distance between the boat and the base of the cliff.

You can check your answers here.

This is the Portfolio icon. Working with Triangles

In Activity 7, you updated your document, Working with Triangles.

Open your document and update previous entries or add new entries for using the primary trigonometric ratios and the Pythagorean Theorem to solve problems.

Consider how the information may be stated in real life situations and how the information can be reflected in the triangle that you use to model the problem.

Remember that you can list each relation more than once in the chart.

Save to your Portfolio

 

This is the dropbox icon. Trigonometry Practice

Complete the worksheet U3A8 Trigonometry 2

 
Consolidation

CONSOLIDATION

This is the Portfolio icon. Mathematical Processes

Complete your entry for the Mathematical Process Connecting in the document U3 Mathematical Processes.

Save to Portfolio.

 

In Unit 2 and Unit 3, you have learned new ideas about triangles that will help you to create models for different situations in the real world. When a situation can be modeled with a right triangle, you have to decide whether you will use the Pythagorean Theorem or a primary trigonometric ratio.

When you use a trigonometric ratio, you need to choose the ratio that ties three pieces of the triangle together -- the reference angle along with 2 side lengths. In creating the solution set-up, you know that you need values for 2 of the three pieces.

 
 

Congratulations, you have completed Unit 3, Activity 8. You may move on to Unit 3, Activity 9.

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